Absolute Value of a Complex Integral

Question

My question may seem naive, but I couldn't find its answer in books, websites, etc.

Assume that I want to calculate numerically the absolute value of the following integral

$$I = \int_0^T\exp(if(t)),$$

where $f(t)$ is a real function of $t$.

Which one of the following is the answer?

1. $\quad|I|^2 = I\cdot I^*$

2. $\quad|I|^2 = \displaystyle\int_0^T\mathrm dt\int_0^t\mathrm dt'\exp(-if(t))\exp(if(t'))$

Any comment or help would be highly appreciated.

Answer 1

The absolute value of any complex number is given by

$$|a|=\sqrt{a \cdot a^*}$$

It does not matter if you get the value of $a$ by integration, differentiation or another operation. Therefore your first solution is correct.

Your second example introduces a dependency between the integrals, which leads to a wrong result.

Answer 2

1) is True and 2) is almost true, following from 1).

\begin{align} |I|^2&=I\cdot I^*\\ &=\left(\int_0^T\exp(if(t))dt\right)\left(\int_0^T\exp(-if(t'))dt'\right)\\ &=\int_0^Tdt\int_0^Tdt'\exp(if(t))\exp(-if(t')) \end{align} So the only difference is that $\int_0^t$ is replaced by $\int_0^T$.